D₅₃×#2231 #27538824

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

D₅₃×#2231

Splittings

Fitting	C₅₃×#2231 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2231

Subgroups

	subgroup	G/H
center	C₂	#476×D₅₃
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2231	C₂
socle	C₂×C₅₃	C₂×#476
prime radical	#135×D₅₃	C₂²
hypercenter	#2231	D₅₃
hypocenter	C₅₃	C₂×#2231
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2231

526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 points:

(1,4)(2,9)(3,15)(5,18)(6,22)(7,20)(8,24)(10,27)(11,30)(12,29)(13,31)(14,19)(16,26)(17,25)(21,28)(23,32)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35);(1,3,14,23,7,16,5,6)(2,8,21,11,12,25,10,13)(4,15,19,32,20,26,18,22)(9,24,28,30,29,17,27,31);(1,2,7,12)(3,13,16,11)(4,17)(5,21,14,10)(6,8,23,25)(9,26)(15,29)(18,31)(19,30)(20,24)(22,27)(28,32);(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)

Conjugacy classes & character table

644 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324300);

D53×#2231 #27538824